Disquisitiones Arithmeticae

In his Preface to the Disquisitiones, Gauss describes the scope of the book as follows: The inquiries which this volume will investigate pertain to that part of Mathematics which concerns itself with integers.Gauss also writes, "When confronting many difficult problems, derivations have been suppressed for the sake of brevity when readers refer to this work."

("Quod, in pluribus quaestionibus difficilibus, demonstrationibus syntheticis usus sum, analysinque per quam erutae sunt suppressi, imprimis brevitatis studio tribuendum est, cui quantum fieri poterat consulere oportebat") The book is divided into seven sections: These sections are subdivided into 366 numbered items, which state a theorem with proof or otherwise develop a remark or thought.

He also realized the importance of the property of unique factorization (assured by the fundamental theorem of arithmetic, first studied by Euclid), which he restates and proves using modern tools.

Ideas unique to that treatise are clear recognition of the importance of the Frobenius morphism, and a version of Hensel's lemma.

The logical structure of the Disquisitiones (theorem statement followed by proof, followed by corollaries) set a standard for later texts.

The Disquisitiones was the starting point for other 19th-century European mathematicians, including Ernst Kummer, Peter Gustav Lejeune Dirichlet and Richard Dedekind.

In section VII, article 358, Gauss proved what can be interpreted as the first nontrivial case of the Riemann hypothesis for curves over finite fields (the Hasse–Weil theorem).